Path-ordering

In theoretical physics, path-ordering is the procedure (or a meta-operator {\mathcal P}) of ordering a product of many operators according to the value of one chosen parameter:

{\mathcal P} \left[O_1(\sigma_1)O_2(\sigma_2)\dots O_N(\sigma_N)\right]:= O_{p_1}(\sigma_{p_1}) O_{p_2}(\sigma_{p_2})\dots O_{p_N}(\sigma_{p_N}).

Here p:\{1,2,\dots ,N\} \to \{1,2,\dots, N\} is a permutation that orders the parameters:

\sigma_{p_1}\leq \sigma_{p_2}\leq \dots \leq \sigma_{p_N}.

For example:

{\mathcal P} \left[ O_1(4) O_2(2) O_3(3) O_4(1) \right]:=O_4(1) O_2(2) O_3(3) O_1(4) .

Examples

If an operator is not simply expressed as a product, but as a function of another operator, we must first perform Taylor expansion of this function. This is the case of the Wilson loop that is defined as a path-ordered exponential; this guarantees that the Wilson loop encodes the holonomy of the gauge connection. The parameter \sigma that determines the ordering is a parameter describing the contour, and because the contour is closed, the Wilson loop must be defined as a trace in order to become gauge-invariant.

Time ordering

In quantum field theory it is useful to take the time-ordered product of operators. This operation is denoted by {\mathcal T}. For two operators A(x) and B(y) that depend on spacetime locations x and y we define:

{\mathcal T} \left[A(x) B(y)\right]�:= \left\{ \begin{matrix} A(x) B(y) & \textrm{ if } & x_0 > y_0 \\ \pm B(y)A(x) & \textrm{ if } & x_0 < y_0. \end{matrix} \right.

Here x_0 and y_0 denote the time-coordinates of the points x and y.

Explicitly we have

{\mathcal T} \left[A(x) B(y)\right]�:= \theta (x_0 - y_0) A(x) B(y) \pm \theta (y_0 - x_0) B(y) A(x),

where \theta denotes the Heaviside step function and the \pm depends on if the operators are bosonic or fermionic in nature. If bosonic, then the + sign is always chosen, if fermionic then the sign will depend on the number of operator interchanges necessary to achieve the proper time ordering. Note that the statistical factors do not enter here.

Since the operators depend on their location in spacetime (i.e. not just time) this time-ordering operation is only coordinate independent if operators at spacelike separated points commute. Note that the time-ordering is usually written with the time argument increasing from right to left.

The S-matrix in quantum field theory is an example of a time-ordered product. The S-matrix, transforming the state at t=-\infty to a state at t=%2B\infty, can also be thought of as a kind of "holonomy", analogous to the Wilson loop. We obtain a time-ordered expression because of the following reason:

We start with this simple formula for the exponential

\exp(h) = \lim_{N\to\infty} \left(1%2B\frac hN\right)^N.

Now consider the discretized evolution operator

S = \dots (1%2Bh_{%2B3})(1%2Bh_{%2B2})(1%2Bh_{%2B1})(1%2Bh_0)(1%2Bh_{-1})(1%2Bh_{-2})\dots

where 1%2Bh_{j} is the evolution operator over an infinitesimal time interval [j\epsilon,(j%2B1)\epsilon]. The higher order terms can be neglected in the limit \epsilon\to 0. The operator h_j is defined by

h_j =\frac{1}{i\hbar} \int_{j\epsilon}^{(j%2B1)\epsilon} dt \int d^3 x \, H(\vec x,t).

Note that the evolution operators over the "past" time intervals appears on the right side of the product. We see that the formula is analogous to the identity above satisfied by the exponential, and we may write

S = {\mathcal T} \exp \left(\sum_{j=-\infty}^\infty h_j\right) = {\mathcal T} \exp \left(\int dt\, d^3 x \, \frac{H(\vec x,t)}{i\hbar}\right).

The only subtlety we had to include was the time-ordering operator {\mathcal T} because the factors in the product defining S above were time-ordered, too (and operators do not commute in general) and the operator {\mathcal T} guarantees that this ordering will be preserved.

See also